This document is a draft and liable to change as I continue the investigation into the ideas and results from from these mental constructs.. you are welcome to comment on the ideas contained and even to use them as long as you acknowledge where they came from.

Dr. Derek Geoffrey Finch Ph.D. BSc. ARCS MInstP. CPhys.


A 'special relativistic invariant' Newtonian gravitational field is developed that results in 'predictions' that match those of the 'classical' tests of Einstein's General theory of relativity but does not give rise to 'singularities' when applied to 'black holes' and which since it is 'linear' should be amenable to 'quantum renormalisation' although this aspect of the theory is not investigated. As an additional benefit to the theory a 'classical' explanation to the shift in electron interference patterns when beams of electrons pass an 'infinitely long' solenoid or magnetic whisker if found as well as a requirement that the gravitational interaction strength must be weaker than that for electromagnetism for all 'normal matter'.


This theory of gravitation was originally developed with the aid of the (now) Dr. Geoff Webb in the early 1980's whilst we were both studying as post graduate students under Dr. Peter Aplin at the University of Bristol as a 'proof' that gravitational radiation would be a necessary consequence of any special realistically invariant theory of gravitation. And whilst many years have passed since we first explored the concepts and arguments that lead to the following work I fully acknowledge the important contribution to these ideas made by Geoffrey Webb those many years ago.


This paper arises out of consideration of a few simple thought experiments into the nature of the gravitational force in a special relativistic universe.  Its conclusions illuminate a number of " problems" in classical general relativity and provide an insight into possible solutions to some of these difficulties.  The approach taken is to start with a classical Newtonian gravitational field and to extend this field view using insights borrowed from electromagnetism and its relativistic extensions by Albert Einstein in order to produce a special relativistic model of gravitation.  This model is then extended using some simple thought experiments and shown to produce results consistent with the classical tests of general relativity but with cosmological insights that point to ways to avoid some of the theoretical difficulties of the classical interpretation of general relativity with respect to gravitational singularities within black holes.  Other interpretation differences are indicated that may allow a consistent quantisation of gravity without the need to invoke higher dimensionality's. 

Derivation of 'special relativistic invariant' Newtonian gravity

Thought experiment 1: the mass/charge balance

The starting point to this derivation is the following thought experiment[1]: Consider a simple balance consisting of a light beam pivoted at some point along its length from which hangs on one side a mass m and on the other a body with a charge q, stood on a massive charged body mass M. and charge minus Q,  Such that the balance is in equilibrium in its rest frame.  Quite clearly if it is in balance in its rest frame then it must be in balance when observed in all other inertial frames.  From classical electromagnetism, or special relativity (the former being the consequence of the latter) we know exactly how the forces between the charges behave and so can directly infer that the gravitational force must also transform in the same way under changes to inertial frames.[2]  The description of the interaction in special relativistic terms involves a Lorenz contraction of the space and time frames in a manner that changes only the structure of the electric field distribution in space and the rate of passage of time, it does not change the source charge of the electrically interacting bodies.  It is simpler to revert to the electromagnetic model and consider the electric field to be independent of the inertial frame with a "correcting" magnetic field existing between the charges.  This view will result in the same conclusions about the electromagnetic forces acting as it is simply a subset of the transformations due to special relativity.  Given that the gravitational and electromagnetic forces must transform in the same way under changes in inertial frame it is reasonable to ask if we can treat the gravitational force (and field) simply has an anti-symmetric (in that like masses attract while like charges repel) version of electromagnetism.  If we take this approach we would have to separate the inertial and gravitational mass transformations under changes in inertial frame and thus abandon the principle of strong equivalence. Which, at first sight, would suggest that any such attempt would be doomed to fail the observational tests associated with the general theory of relativity, but would, at the same time, result in a "linear theory" that should be easier to fit with quantum theory.  Applying this simple anti-symmetrical approach would imply the existence of the gravitational analogue to the magnetic field by which masses moving at the same velocity and side-by-side relative to an observer would experience a repulsive force (note this is in the opposite direction to the electromagnetic case) that would just counter the increased attraction due to the Lorenz contraction of space with its concomitant increase in field density at right angles to the direction of motion of the bodies.  (If a model in which the gravitational mass of the bodies increase is in line with the inertial mass is assumed, then this repulsive force must rise as g3 where g=(1/)), not just g, as both bodies mass will increase by g as well as the field between them rising by g.) If we now consider the effect of stopping one of these masses and remembering that the gravitational field is constrained to move at the speed of light (or less), or alternatively we apply the transformations from the Maxwell equations between changing magnetic and electric fields to the dynamic and static gravitational fields we find that slowing one of the bodies will also "slow" the other[3] (unlike in electromagnetism where slowing one of a pair of like charges would have increased the speed of the other). So if the dynamic force transformed as g3 as the bodies approached the speed of light their 'inertial' mass would approach zero as this mutual induction force which reduces the effective inertial mass exceeds the increase by g in inertial mass that occurs due to the special relativistic mass increase. This is counter to observation of high-energy cosmic rays or highly relativistic particle accelerator results. 

Thought experiment 2: Like charges repel, unlike attract as consequence of conservation of energy.

To proceed further with our exploration of special relativistic classical Newtonian gravitation we will have to divert into another set of thought experiments.  This time into a set that are, in all probability, outside the limits of practicality as actual experiments but which will be illuminating in their conclusions.  First let us consider the nature of "force" from a classical (as opposed to quantum) special relativistic point of view.  We observe that the conservation of energy is axiomatic within the formulation not only of special relativity but most of modern physics so considering two charges: q1 and q2 separated by a distance r and initially both stationary in some rest frame, we observe that each of these charges is surrounded by an electric field. Now the energy contained in that field is given by half the integral over all space of the square of the sum of the fields times the permittivity of the space.  Hence if the bodies move relative to one another the energy contained in the fields changes.  Since energy is conserved, the bodies either cannot move or must move in a fashion such as to conserve energy.  In this simple universe, with just two charges, the only place the energy can go is into the 'kinetic' energy of the bodies.  Indeed if they start to move a magnetic field will come into existence which will also contain energy.  In order to conserve energy, if the charges are alike and the bodies start to move apart, since the energy in the electric field will decrease, that in the magnetic field must increase and the bodies must move faster moving them further apart; which will require a further compensatory increase in velocity: i.e. they will accelerate apart, likewise if the charges involve opposite charges the bodies must accelerate towards one another in order to conserve energy.  This indicates that force is a consequence of the conservation of the field energies.  An interesting aside is to note that this provides a classical exploration of the deflection and electrons passing an infinitely long solenoid, an experiment which is usually claimed as proof of the existence of the quantum vector and scalar potentials.  The explanation of this experiment according to this paper's model of force is that as the electron passes to one side of the solenoid the magnetic field of the electron that intersects the solenoid will either be reinforced or decreased depending on the direction of the field in the solenoid and the side that the electron passes.  If it is reinforced then the energy in the field in that part of the solenoid will be increased, so the energy in the rest of the electromagnetic field must decrease in order to conserve energy and the electron must slow down, conversely if it passes on the other side of the solenoid the field in the solenoid will oppose will oppose the field from the electron decreasing the energy in that part of the field necessitating an increase in the electrons speed as it approaches the solenoid. Having passed the solenoid these effects are reversed and the electrons would continue with their original speed.  The combined effect would be to introduce an advancing position of the electrons passing on one side of the solenoid and a retardation in position of electrons passing on the other i.e. a change in the relative phase of the electrons as observed experimentally.

To resume our original argument we have now shown that like charges must repel and unlike attract one another as that is the only way the field energies can be conserved.  This, at first sight, provides an insurmountable problem with the theory of gravitation that we have been considering, as a moment's thought will show that even if (as it must be) the energy contained in the gravitational field is negative then, since the energy in the dynamic gravitational field must also be negative, like masses must repel and, if they exist, unlike attract.  This is clearly in contradiction to observation.  Since this approach has been a simple exploration of the logical consequences of some basic principles then either at least one of these principles is in error or we are missing some vital fact in reaching our conclusions.  A careful consideration of the observation that like masses attract gives a way out of this conundrum.  All the masses we observe being attracted to one another are not just simple masses but rather composed of tightly packed charged bodies (even neutrons are composed of charged quarks) which changes the situation as now there are three fields that are exchanging energy, the static gravitational field, the dynamic gravitational field and the internal magnetic fields surrounding the charges that make up "normal matter" (the electric fields due to these charges is ignored at this juncture as they are independent in velocity and would only come into play if the bodies were distorted as in a collision).  Now if the negative energy contained in the gravitational field is lower in magnitude than the positive energy contained in the magnetic fields, then, if the bodies are moving towards one another, the energy in the static gravitational field will become more negative and so the combined effect of the dynamic gravitational field energy and the positive magnetic field energy must be to increase (become more positive).  Since by definition the positive energy in the magnetic field exceeds the magnitude of the negative energy in the dynamic gravitational field the bodies must increase in speed: i.e. accelerate towards one another.  This provides an immediate consequence which is that whichever of the two fields is weaker (has the lower energy) in any situation it will act such that like charges, or masses, attract whilst unlike repel and whichever is stronger will have the property that likes repel while un-likes attract.  Thus explaining why gravity is much weaker than electromagnetism, since where it 'stronger' we would simply swap our view of them.  An additional consequence is that when a large amount of mass is pushed together a point will be reached when the energy in the gravitational field will equal or exceed that in the electromagnetic field (since the energy in the gravitational field will rise as the square of the mass and that in the electromagnetic field only linearly) at this point the masses will start to repel instead of attract.  (It can be shown that this will occur when the bodies reach the General relativistic event horizon. ) That is the singularity predicted by General relativity will not exist.  Further consequences for black holes will be explored later in this paper as will be general relativistic interpretation of the same properties.  Since the inertial mass of gravitationally bound body is less than the sum of the inertial masses of its constituent parts this has an effect on planetary orbits and on the frequency of light emitted in various interactions.  Most notably it produces a procession of perihelion that matches the predictions of general relativity and an apparent gravitational red shift that also matches the predictions of general relativity.  The former can be shown by considering the increase in inertial mass due to the kinetic energy (primarily the energy contained in the magnetic fields and their equivalents for the strong and weak nuclear forces); the decrease in inertial mass due to the mutual interaction of the gravitational equivalent of the magnetic field, equivalent to the change in potential energy of the now bound body and finally the increase in gravitational field linking the planet and parent star due to their motion relative to an observer in their centre of gravity frame of reference.  For a near circular orbit the kinetic energy term is half that of the potential energy term and the increase in mutual force due to the Lorenz contraction of space produces an extra attraction that cancels the increase in inertial mass due to the body's kinetic energy. Giving rise to a pertebation term the same as that obtained from General relativity.

When considering the gravitational red shift of emitted photons it is worth remembering that special relativity forbids any force from acting on a photon in the direction of the photons motion other than when the photon is emitted or absorbed.  So according to General relativity, which is fully compatible with special relativity, the gravitational red shift as photons climb out of a gravitational potential well occurs not because they are acted on by the gravitational force but rather that the emitting body emits them at a lower frequency than would have been the case had they not been at the bottom of the potential well (clocks run slower at the bottom of a gravitational potential well than the top) on the other hand the view of gravity in this paper states that the photons were emitted with a longer wavelength, that is at a lower frequency, not because clocks are running slower but rather that the interaction occurs with lower inertial masses and so emits photons with lower energies. 

Similarly the bending of light as it passes a massive body, e.g. the Sun, by twice the Newtonian angle is simply a consequence of the fact that according to special relativity forces can only act at right angles to the direction of travel for photons.

It is worth at this point investigating the question of what happens when a large massive body collapse under its own weight: According to the classical view of general relativity, what happens is that the material outside the Schwarzschild radius is seen to collapse ever slower on to the event horizon as time slows down for that material as seen by an outside observer, in fact although the material reaches the speed of light as it approaches the event horizon, the speed of light as seen by the outside observer approaches zero. Material inside the Schwarzschild radius cannot of course be seen at all by the outside observer but is supposed to be falling in faster than the speed of light. This motion is caused by the curvature of 'space time' that is not by a curved space warping just distances but by changes to the rate of passage of time as well. In contrast to this picture the behaviour envisaged in this paper is that as the material making the body becomes ever more dense so the positive energy stored in the electric field between its constituent parts becomes lower, the negative gravitational energy in its static gravitational field grows (more negative) and so the energy stored in its magnetic field must grow causing it to accelerate inwards, but as the density rises to the point in which the magnitude of the negative gravitational energy equals the positive electromagnetic energy two connected things occur, firstly the inertial mass of the constituent parts approaches zero and the constituent parts of the body will rapidly accelerate to the speed of light, this results not only in an accelerating collapse but also in the specific heat of the matter approaching zero, thus the bodies temperature will rise until pair creation limits that rise, i.e. black holes are very hot, at least at first, and will radiate strongly but will equally cool down very fast as they have a low specific heat. As the collapse proceeds further and the magnitude of the energy in the gravitational field exceeds that in the electromagnetic field a phase change occurs and like masses repel whilst like charges attract and unlike charges repel. The full picture of behaviour will also depend on the way the energy is held in the strong and weak nuclear forces and is beyond the scope of this paper. In this picture there is no singularity or any breakdown in 'physics' inside the 'event horizon'. This mimics the view that would occur if within the event horizon in general relativity if we accept that bodies travel faster than the speed of light have a time reversal, since under time reversal attraction becomes repulsion and vice a versa.

On a more speculative note since this model of gravitation allows for negative gravitational mass we can speculate that particles with such mass would have been produced in 'the big bang' in which case we would have to ask where they were. Since negative mass is repelled by positive mass we would not expect to find any on earth, or indeed in the local galactic cluster as it would have fallen outwards at a great acceleration long ago unless it was newly created or fired at us with great energy. So if such negative mass does exist (it would have positive inertial mass as the latter is just a consequence of its electromagnetic, strong and weak nuclear self induction.) it should primarily form an intermeshing network of galactic super clusters as the early universe would be like a mixture of two immiscible fluids of equal density and so would start to separate into a network of 'blobs' That we cannot see such a interlacing network would suggest that either such mass is invisible (dose not interact with light), that we are misinterpreting our observations or that it dose not exist. Though on a more speculative note still if 'anti matter' was also of negative gravitational mass it might explain why we appear to live in a universe predominated by matter. Indeed from a theoretical view point it would be neat if the total amount of gravitational mass in the universe was zero since then the formation of the universe would still conserve the total gravitational mass and there would be no need in an open universe (one that is flat or has a positive curvature) to violate the special relativistic ban on fields propagating at the speed of light in the direction of the field vector (I.e. there are no longitudinal light waves). Indeed if the total amount of (negative) gravitational energy in the universe was also equal to the total positive energy in the other fields so that the total energy in the universe was zero there would be no problem with creating a 'big bang' since energy would be conserved, at zero, for all time and such a universe would overall show itself to be 'flat' and not 'slow down' in its expansion. It would also, as a quantum fluctuation, be 'infinitely' long lasting as its life time times its total energy uncertainty would still be h.

The one place that such negative gravitational mass might show itself would be in high energy cosmic rays and it would be worth looking at the (inertial) mass/velocity distribution of these very energetic particles to see if any show the effect of being decelerated by the gravitational field of our local galactic cluster.


On the principal of strong equivalence

The gravitational theory proposed in this paper clearly 'violates' the principal of strong equivalence under changes to inertial frames and as such in order to be taken seriously it must be shown that such a violation is both sensible and more importantly is not in violation of  observation. Firstly to see that it is logically valid we need consider some simple thought experiments, the first is of course the balance thought experiment above but a second related one is a modification of an old special relativistic 'paradox', that of a rod sliding over a table with a hole in it. In the original 'paradox' a rod of length l slides over a table with a hole of width l in it at a relativistic speed and it is argued that in the rods frame as the hole is now less than l wide the rod crosses the hole whilst in the tables frame the rod is shorter than the hole and so falls through. The way out of the paradox is that in both frames as soon as the front of the rod passes the edge of the hole it starts to drop and so in both frames fails to cross the hole. But a small pair of modifications to the scenario reintroduces the problem the changes are to 'bend' the rod such that in the rest frame of the planet the only place the table exerts any upward force on the rod is at its centre and the second modification is that instead of the rod sliding over the table the table slides under the rod i.e. the rod remains stationary in the rest frame of the planet it is on. Now in the rod's frame its tip is over the far side of the hole before the centre of the rod passes over the hole so never drops below the edge of the hole and whilst it might have a 'bumpy' ride it passes over the hole but in the tables frame not only is the rod shorter but both it and the planets mass have increased with the result that the rod will experience a much greater gravitational force pulling down on its whole length and its tip will need support from the table and will start to fall as soon as it passes over the holes edge. Now this is of course nonsense and so one of three things must be true, either the rod's stiffness increases like gamma squared (increase in planets mass by gamma, rods mass by gamma and field strength due to Lorenz contraction by gamma and decrease in force due to time dilation)  or there is some countering gravitational dynamic force that decreases the force due to gravity from co-moving objects by gamma cubed (i.e. an additional gamma squared factor beyond the time dilation effect) or that the source masses remain constant and the gamma increase in field strength is countered by the time dilation effect. The first of these can be ruled out by re thinking the experiment using an electrically charged 'bent' rod on an electrically charged 'planet' since we know in this case that the only change in force is the increase by gamma of the field strength countered by the time dilation effect that we commonly think of as the effect of a magnetic field. Hence since in this latter case there is no increase in 'stiffness' there cannot be any in the former case. And in the second case we would observe a significant effect on ultra-relativistic particles that is not observed. From an observational perspective we must explain why if the principal of strong equivalence is not true we find that the result of the Eotvos and its latter incarnations show no difference in the ratio of inertial to gravitational mass of stationary objects made from differing materials as not only will the numbers and ratios of various subatomic particles that make up the bodies differ so will the instantaneous velocities of the constituent components. That the bodies are made of essentially the same constituents (leptons and quarks along with the photons and gluons that hold them together) may be the explanation, certainly the differing numbers of virtual positrons and electrons in the differing bodies can be allowed for by recognising that the gravitational mass of a virtual pair created from the vacuum must be zero (consider a box enclosing a vacuum, the internal gravitational mass must be zero as the integral of the normal of the gravitational field though the surfaces of the box is zero yet the box is filled with virtual pairs popping into and out of existence all the time). But I recognise that this explanation of the Eotvos results is not elegant and leaves something to be desired. I leave the explanation to others who might have better insight into this than I.


On the negative energy in a gravitational field

The concept of an absolute negative energy is, to many, problematic and indeed in one way is not necessary in that if energy is conserved we can add or subtract an arbitrary constant energy to any closed system. And indeed we can also just scale the energy in such a closed system by any arbitrary positive or negative non-zero constant without changing the dynamics of the system. So what is meant by stating that the energy contained in the gravitational field is negative is that energy can be extracted from a gravitational field by increasing the magnitude of the integral over all space of the square of field strength and conversely to reduce the magnitude of that integral we need to put energy into the system. To see that this is the case, consider a spherical shell of uniform mass density and radius R made of non-electrically charged particles that only interact via the gravitational field initially at rest. Each of the particles is attached to a non massive string to a dynamo so that if they move inwards they produce a current that charges a battery. Clearly as they 'fall' in they will transfer energy from the gravitational potential energy to the chemical energy in the batteries. Where then dose that gravitational potential energy come from? If we where considering an electromagnetic equivalent situation (where the charges would fly apart) we would identify the energy as having come from the electric field, so in the gravitational case we should also do so. In the electromagnetic case energy is removed from the field as the spheres radius increases, this increase in radius increases the volume inside the spherical shell where the field is zero at the expense of the region outside the shell where it is non-zero hence decreasing the integral over all space of the square of the field. But in the gravitational case the radius of the spherical shell decreases so reducing the volume with zero field and increasing the region with a field present that is increasing the integral over all space of the square of the field. The obvious conclusion is that the energy contained in a gravitational field is 'negative' in that the larger the integral the more energy has been 'removed' from the field and defining the energy of empty space to be zero and the energy contained in an electromagnetic field to be positive we have to conclude that the energy contained in a gravitational field is negative.

As mentioned previously if we use a nonzero (very large) value for the energy content of 'empty space' then we could avoid saying that the energy in a gravitational field is negative by simply adding this large positive constant energy to all our field energy calculations but to do so is to introduce an unnecessary complication to our thinking and calculations and invoking Occam's razor we should simply accept that the energy contained in the gravitational field is negative.

Note, the old school boy/girl adage that things always move in a fashion so as to minimise there energy is not true, they always move so as to conserve their energy and hence it is the balance between the energies in the various fields (electric, magnetic, static gravitational, dynamic gravitational etc.) that determine the dynamics of a system.

On gravity and quantum mechanics

This paper makes no attempt at considering the quantum nature of gravity, but since it is a linear theory in which time maintains its absolute sense and forces are mediated by fields that do not act as their own sources it is hoped that it is more amenable to being cast as a quantum field theory along the lines of QED than current attempts at a quantum gravity that fits with the curved space time of general relativity.


On cosmology

It would be nice, though not essential, if anti-matter was shown to be of negative gravitational mass and that this was the explanation as to why matter is seen to dominate the universe, since then all the anti-matter would have 'fallen' out of our local galactic super cluster and indeed would have provided a mechanism by which all the matter and anti-matter in the early universe would not have annihilated themselves in an orgy of mutual destruction but at present this dose not seem to be the case.

Equally there are some observational issues around the dynamics of very massive stars/black holes which I am not happy that I understand well enough to decide if the observational evidence supports or eliminates this explanation of gravity, in particular is the question of inertial vs. gravitational mass for such bodies, though General relativity would have similar problems of stellar dynamics based on gravitational time dilation rather than changes to the inertial to gravitational mass ratio.


On the relationship with the General theory of relativity

The origin of this work was not to replace General relativity but to look at all special relativistic valid gravitational theories of gravity in order to determine if all such theories had, of necessity to result in accelerating masses producing some form of gravitational radiation, which it has done. But beyond that it has provided another way of looking at gravitation in a 'flat space time' as opposed to general relativities curved space time. Where in the General theory of relativity gravity acts by changing the geometry of space-time, that is by changing distances and the rate at which time passes, this field approach changes the inertial mass of bodies and in doing so reproduces the observational 'test' results of General relativity. This then leads to the question as to whether the two are the same with only a change in 'viewpoint'. Are the theories isomorphic? Both start from the same initial assumption of the validity of the Special theory of relativity, both start from a set of hidden assumptions about the conservation of energy and of Newtonian gravity in the non-relativistic, 'low mass' limit so given that neither make a logical error in their subsequent reasoning then they indeed should be isomorphic and where their conclusions seem to differ it should be possible with the right transform to reconcile the differences. In particular the elimination of the troublesome singularity at the centre of 'black holes' by the recognition that within the Schwarzschild radius time reversal will result in repulsion from the centre rather than continued collapse and hence no singularity that maps to the reversal of the role played by gravity and electromagnetism due in effect to a reversal of inertial mass. Which is the 'correct' view to some extent depends on one's definition of time; the current definition based on the 'time taken' for a number cycles from a given atomic transition would match that used by General relativity  in that the rate of passage of time will differ depending on one's relative gravitational potential. If a definition of time was adopted that was not dependant on the gravitational potential, as assumed in the definition of time used in Quantum mechanics then the better model would appear to be this new form.

Areas of uncertainty and outstanding issues

There are a number of 'niggles' and issues that this work to date leaves unresolved, none are show stoppers at present but really could do with resolving, they are, in no particular order:

The expansion of the universe

This theory allows for a 'big bang' and for a continuing expansion and if negative gravitational mass exists for an expansion that continues unabated but not for an accelerating expansion as now appears to be observed. Nor dose it provide the extra red shift mechanism provided by the slower running of clocks at earlier epochs that General relativity provides due to the expansion of space-time that allows.

Photon mass

The observational evidence is limited, but what we do have suggests that photons have positive gravitational mass this either implies that negative gravitational mass dose not exist or that we have missed the light gravitationally bent 'the other way'. Though if we cannot 'see' negative gravitational mass photons and that is what is emitted by bodies with negative gravitational mass that might explain why we do not see the expected interpenetrating negative mass galactic clusters that should 'fill' the voids between the normal mass clusters if negative gravitational mass exists, but then we should find 'missing energy' in results from high energy particle collisions.

 Further that photons carry gravitational mass might imply by symmetry that gravitons should carry charge, but they do not.


Do neutrinos carry gravitational mass? And if so is it only positive?


[1]  It is in fact a common experiment in essence; as anyone who has applied a charge to a set of capacitor plates in a 'gold leaf electroscope' and seen the electrostatic force balanced by the gravitational force acting on the gold leaf from differing inertial frames has carried it out in essence.

[2]  This argument must also apply to the other fundamental forces known to physics as the charges, or masses, could in principle be replaced by bodies interacting by the strong or weak nuclear forces.

[3]  This is consistent with the predictions of special relativity as it implies a reduction in the inertial mass of the two parallel moving bodies due to their mutual interaction.  This reduction in inertial mass can be understood from the observation that the gravitational potential energy of the system is lower when the two masses are bought together than when they are far apart. Hence the inertial mass must be lower when they are closer together.